Hopf bifurcation analysis of reaction–diffusion neural oscillator system with excitatory-to-inhibitory connection and time delay

The active control approach generally requires power input to suppress vibrations of structures, while the conventional passive manner often causes waste of energy after transferring vibrations of the primary structure to the auxiliary system. In this work, an innovative control strategy based on energy harvesting for efficiently suppressing the cross-flow-induced vibrations such as galloping is proposed. The novel design facilitates the harvester of not only alleviating the oscillation of the primary structure but also seizing the transferred vibrational energy. An analytical model for the coupled nonlinear dynamical system is established by utilizing the Euler–Lagrange principle and implementing the Galerkin discretization. The impacts of the electrical load resistance and tip mass of the energy harvester on the coupled frequency, damping, and the onset speed of instability of the coupled multi-mode system are investigated in details. The results show that there exists an optimal load resistance for each tip mass which maximizes the onset speed of galloping. For control purposes, it is found that there is a well-defined tip mass of the energy harvester at which the coupled system has the highest onset speed of instability, and hence, the bluff body has the lowest vibration amplitude for all considered load resistances. However, to efficiently harvest energy and control the bluff body, both the tip mass of the energy harvester and electrical load resistance can be accurately determined.     

Hopf bifurcations through delay in pilot reaction in a longitudinal flight

The paper is devoted to the study of pilot induced oscillations in the landing transition between the approach task and flare to touch-down. These oscillations are proved to appear in a longitudinal flight model when the delay in pilot’s reactions exceeds a certain threshold for which the stability of equilibria is lost and a Hopf bifurcation appears. The formulae needed to compute the Lyapunov coefficient and an approximation of the solution are developed for the delay differential equations that model the pilot–vehicle interaction in landing task. These are applied for a concrete model.     

A reaction–diffusion model for competing languages

The extinction of many of the world’s minority languages is of great concern as language death can lead to the irrevocable loss of cultural information. This often occurs through a process of language shift, where individuals switch from speaking one language to a different, more dominant, language. To prevent the loss of language, it is necessary to determine whether language loss is inevitable or if languages can coexist. We address this question by constructing a nonlinear system of reaction–diffusion equations to model the spread of two competing languages, u and v, which vary temporally and spatially. Language u is assumed to confer a relative status advantage to its speakers, thus individuals may convert from language v to language u. The four constant system equilibria are found. Instability and stability conditions are found for each equilibrium. We conclude that the coexistence of both languages u and v is globally stable, subject to certain constraints on the growth rate of each language and the initial values of both u and v.     

Stability and global Hopf bifurcation in toxic phytoplankton–zooplankton model with delay and selective harvesting

Consider that some zooplankton can be harvested for food and some phytoplankton can liberate toxin; a toxin producing phytoplankton–zooplankton model with delay and selective harvesting is proposed and investigated. We discuss the stability of equilibria and perform the analysis of Hopf bifurcation. More precisely, the global asymptotical stability of equilibria is investigated by the Lyapunov method and Dulac theorem. In addition, the computing formulas of stability and direction of the Hopf bifurcating periodic solutions are also given. Furthermore, we prove that there exists at least one positive periodic solution as a time delay varies in some regions by using the global Hopf-bifurcation result of Wu (Trans. Am. Math. Soc. 350:4799–4838, 1998) for functional differential equations. Finally, the impact of harvesting is discussed along with numerical results to provide some support to the analytical findings.     

Hopf and Bogdanov�CTakens bifurcations in a coupled FitzHugh�CNagumo neural system with delay

In this paper, Hopf bifurcation and Bogdanov?CTakens bifurcation with codimension 2 in a coupled FitzHugh?CNagumo neural system with gap junction are investigated. At first, a general bifurcation diagram on the plane of coupling strength and delay is derived. Then, explicit algorithms due to Hassard and Faria are applied to determine the normal forms of Hopf and Bogdanov?CTakens bifurcations, respectively. Next, we analyze the codimension-2 unfolding for Bogdanov?CTakens bifurcation, and give complete bifurcation diagrams and phase portraits. Furthermore, we also consider the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. By the results of theoretical analysis, we obtain that the values of coupled strength, which make the transmission and received signals be synchronous and anti-phase, are opposite. And universal unfolding of Bogdanov?CTakens bifurcation indicates that the neuron signals can transit between resting and spiking.     

Hopf bifurcations on invariant manifolds of a modified Fitzhugh–Nagumo model

Nonlinear Dynamics - In this paper, we investigate the nonlinear vibration of a metamaterial structure that consists of a rotating cantilever beam attached to a periodic array of...     

Coherence resonance in an autaptic Hodgkin–Huxley neuron with time delay

Autapses, synapses between a neuron and itself, are believed to serve as a regulator of information processing in the nervous system. Noise, as random or unpredictable fluctuations, affects nearly all aspects of nervous function. In this work, we studied the regulatory ability of an autapse on firing behaviors of a Hodgkin–Huxley neuron in response to synaptic-like signals in the presence of noise. For weak subthreshold synaptic input, the autaptic neuron produced the similar firing behavior governed by the noise in case of either determinate or random synaptic input. The critical noise intensity necessary to trigger the spike train slightly was decreased with determinate low-frequency input. Under a strong suprathreshold synaptic current, noise-induced frequency resonance and precise responses mostly occured with a high-frequency determinate input. With a random strong synaptic input, an increase in the autaptic delay led to a resonance-like response, but had no observable effects in cases with a short delay time under a low-frequency input.     

Adaptive synchronization of stochastic neural networks with mixed time delays and reaction–diffusion terms

In this paper, the problem of exponential synchronization is investigated for a class of stochastic perturbed chaotic neural networks with both mixed time delays and reaction?Cdiffusion terms. By employing Lyapunov?CKrasovskii functional and stochastic analysis approaches, an adaptive controller is designed to guarantee the exponential synchronization of proposed neural networks in the mean square. In particular, the mixed time delays in this paper synchronously consist of constant delay in the leakage term (i.e., ??leakage delay??), discrete time-varying delay and distributed time-varying delay which are more general than those discussed in the previous literature. Furthermore, our synchronization criteria are easily verified and do not need to solve any linear matrix inequality. Therefore, the results obtained in this paper generalize and improve those given in the previous literature. Finally, the extensive simulations are performed to show the effectiveness and feasibility of the obtained method.     

On the nonlinear dynamics of an ecoepidemic reaction–diffusion model

A reaction–diffusion ecoepidemic model of predator–prey type with a transmissible disease spreading among the predator species only is considered. The longtime behavior of solutions is analyzed and, in particular, absorbing sets in the phase space are determined. Conditions guaranteeing the non existence of non-constant equilibria have been found. Linear and non-linear stability conditions for biologically meaningful equilibria are determined.     

Dynamics of solid-melt front migration from a reaction–diffusion model

Using physical experiments we investigated the evolution of thermally driven melt patterns in a semi-infinite solid crystalline phase subjected to uniform heating from one side, maintaining melting temperature. We treat the melt initiation phenomenon theoretically in the perspective of two-phase interactions on the microscopic level, and propose a new reaction–diffusion model based on the prey–predator dynamics. This model predicts the fractal behavior of melt fronts observed in the experiments.     

Wave features of a hyperbolic reaction–diffusion model for Chemotaxis

The Extended Thermodynamic theory is used to derive a hyperbolic reaction–diffusion model for Chemotaxis. Linear stability analysis is performed to study the nature of the equilibrium states against uniform and nonuniform perturbations. A particular emphasis is given to the occurrence of the Turing bifurcation. The existence of traveling wave solutions connecting the two steady states is investigated and the governing equations are numerically integrated to validate the analytical results. The propagation of plane harmonic waves is analyzed and the stability regions in terms of the model parameters are shown. The frequency dependence of the phase velocity and of the attenuation is also illustrated. Finally, in order to have a measure of the non linear stability, the propagation of acceleration waves is studied, the wave amplitude is derived and the critical time is evaluated.     

Dynamical stability in a delayed neural network with reaction–diffusion and coupling

Nonlinear Dynamics - In this paper, a delayed neural network with reaction–diffusion and coupling is considered. The network consists of two sub-networks each with two neurons. In the first...     

Hopf bifurcation for a small-world network model with?parameters delay feedback control

To delay the onset of undesirable bifurcation, the bifurcation control has become a subject of intense research activities. In this paper, a small-world network model with the delay feedback is considered, in which the strength of feedback control is a nonlinear function of delay. With this controller, one can change the critical value of bifurcation, and thus enlarge the stable region. Moreover, by adding some proper slowly varying parts into the bifurcation parameters, the stability can be improved. Numerical results show that the dynamics of the small-world network model with the controller of delay-dependent parameters is quite different from that of a system with the controller of delay-independent parameters only.     

Bifurcations and chaos in a discrete predator–prey model with Crowley–Martin functional response

In this paper, a discrete-time predator–prey model with Crowley–Martin functional response is investigated based on the center manifold theorem and bifurcation theory. It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation. An explicit approximate expression of the invariant curve, caused by Neimark–Sacker bifurcation, is given. The fractal dimension of a strange attractor and Feigenbaum’s constant of the model are calculated. Moreover, numerical simulations using AUTO and MATLAB are presented to support theoretical results, such as a cascade of period doubling with period-2, 4, 6, 8, 16, 32 orbits, period-10, 20, 19, 38 orbits, invariant curves, codimension-2 bifurcation and chaotic attractor. Chaos in the sense of Marotto is also proved by both analytical and numerical methods. Analyses are displayed to illustrate the effect of magnitude of interference among predators on dynamic behaviors of this model. Further the chaotic orbit is controlled to be a fixed point by using feedback control method.